| L(s) = 1 | + (3.06 − 2.57i)2-s + (17.0 + 14.2i)3-s + (2.77 − 15.7i)4-s + (54.6 − 19.8i)5-s + 88.9·6-s + (49.4 − 18.0i)7-s + (−32.0 − 55.4i)8-s + (43.6 + 247. i)9-s + (116. − 201. i)10-s + (−100. − 174. i)11-s + (272. − 228. i)12-s + (−25.2 + 142. i)13-s + (105. − 182. i)14-s + (1.21e3 + 441. i)15-s + (−240. − 87.5i)16-s + (152. + 865. i)17-s + ⋯ |
| L(s) = 1 | + (0.541 − 0.454i)2-s + (1.09 + 0.916i)3-s + (0.0868 − 0.492i)4-s + (0.977 − 0.355i)5-s + 1.00·6-s + (0.381 − 0.138i)7-s + (−0.176 − 0.306i)8-s + (0.179 + 1.01i)9-s + (0.367 − 0.636i)10-s + (−0.251 − 0.434i)11-s + (0.546 − 0.458i)12-s + (−0.0413 + 0.234i)13-s + (0.143 − 0.248i)14-s + (1.39 + 0.507i)15-s + (−0.234 − 0.0855i)16-s + (0.128 + 0.726i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 + 0.161i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.986 + 0.161i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.81524 - 0.309562i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.81524 - 0.309562i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-3.06 + 2.57i)T \) |
| 37 | \( 1 + (-2.22e3 + 8.02e3i)T \) |
| good | 3 | \( 1 + (-17.0 - 14.2i)T + (42.1 + 239. i)T^{2} \) |
| 5 | \( 1 + (-54.6 + 19.8i)T + (2.39e3 - 2.00e3i)T^{2} \) |
| 7 | \( 1 + (-49.4 + 18.0i)T + (1.28e4 - 1.08e4i)T^{2} \) |
| 11 | \( 1 + (100. + 174. i)T + (-8.05e4 + 1.39e5i)T^{2} \) |
| 13 | \( 1 + (25.2 - 142. i)T + (-3.48e5 - 1.26e5i)T^{2} \) |
| 17 | \( 1 + (-152. - 865. i)T + (-1.33e6 + 4.85e5i)T^{2} \) |
| 19 | \( 1 + (-518. - 434. i)T + (4.29e5 + 2.43e6i)T^{2} \) |
| 23 | \( 1 + (179. - 310. i)T + (-3.21e6 - 5.57e6i)T^{2} \) |
| 29 | \( 1 + (1.38e3 + 2.40e3i)T + (-1.02e7 + 1.77e7i)T^{2} \) |
| 31 | \( 1 + 44.3T + 2.86e7T^{2} \) |
| 41 | \( 1 + (664. - 3.76e3i)T + (-1.08e8 - 3.96e7i)T^{2} \) |
| 43 | \( 1 + 1.60e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (1.31e4 - 2.27e4i)T + (-1.14e8 - 1.98e8i)T^{2} \) |
| 53 | \( 1 + (-9.16e3 - 3.33e3i)T + (3.20e8 + 2.68e8i)T^{2} \) |
| 59 | \( 1 + (4.23e4 + 1.54e4i)T + (5.47e8 + 4.59e8i)T^{2} \) |
| 61 | \( 1 + (-5.64e3 + 3.20e4i)T + (-7.93e8 - 2.88e8i)T^{2} \) |
| 67 | \( 1 + (-6.66e3 + 2.42e3i)T + (1.03e9 - 8.67e8i)T^{2} \) |
| 71 | \( 1 + (-1.60e4 - 1.34e4i)T + (3.13e8 + 1.77e9i)T^{2} \) |
| 73 | \( 1 + 3.84e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + (8.34e3 - 3.03e3i)T + (2.35e9 - 1.97e9i)T^{2} \) |
| 83 | \( 1 + (2.51e3 + 1.42e4i)T + (-3.70e9 + 1.34e9i)T^{2} \) |
| 89 | \( 1 + (1.24e4 + 4.52e3i)T + (4.27e9 + 3.58e9i)T^{2} \) |
| 97 | \( 1 + (-1.10e4 + 1.91e4i)T + (-4.29e9 - 7.43e9i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.77923585038299841553749855082, −12.77632147679727732311387070367, −11.18542548141985693956069195920, −10.02106414734593101752945293334, −9.306450997696193837853853247957, −8.086182504214237967230980199428, −5.93458734562155046691245136528, −4.60538221095666757155251458440, −3.28642367006536806623851506618, −1.80175938198975545469504307556,
1.87307916962715302827335343486, 3.01823093044618271298465581634, 5.15888299298152607644015400967, 6.64361257142761727475481348770, 7.61956683463167145157101518194, 8.742304137145812283642760875178, 10.04508241962696527381861308695, 11.78499420473956560094290860925, 13.05786749730959653906031057791, 13.65500905658864735403307883294
